12/03/2019
To make statistical problems more tractable:
\(\cdot\) Common to pool data eg. spatially
\(\cdot\) Partition a region eg. consider region by region
National Resource Management Regions
CSIRO and Bureau of Meteorology, 2015. Climate change in Australia information for Australia's natural resource management regions: Technical report.
Whan, Kirien, and Maurice Schmeits. "Comparing area probability forecasts of (extreme) local precipitation using parametric and machine learning statistical postprocessing methods." Monthly Weather Review 146.11 (2018): 3651-3673.
How should partition regions for the analysis of extremes?
Create regions that are likely to experience similar impacts
These regions can then inform our statistical analysis
1. Regionalisation
2. Visualise spatial dependence
Require: Measure of closeness between two locations
Want: Form clusters based on extremal dependence
Solution: The F-madogram distance
Bernard, Elsa, et al. "Clustering of maxima: Spatial dependencies among heavy rainfall in France." Journal of Climate 26.20 (2013): 7929-7937.
\[d(x_i, x_j) = \tfrac{1}{2} \mathbb{E} \left[ \left| F_i(M_{x_i}) - F_j(M_{x_j})) \right| \right]\] where \(M_{x_i}\) is the annual maximum rainfall at location \(x_i \in \mathbb{R}^2\) and \(F_i\) is the distribution function of \(M_{x_i}\).
Advantages:
Cooley, D., Naveau, P. and Poncet, P., 2006. Variograms for spatial max-stable random fields. In Dependence in probability and statistics (pp. 373-390). Springer, New York, NY.
For \(M_{x_i}\) and \(M_{x_j}\) with common GEV marginals is \[\mathbb{P}\left( M_{x_i} \leq z, M_{x_j} \leq z \right) = \left[\mathbb{P}(M_{x_i}\leq z)\mathbb{P}(M_{x_i}\leq z)) \right]^{\tfrac{1}{2}\theta(x_i - x_j)}. %= \exp\left(\dfrac{-\theta(h)}{z}\right),\] where \(\theta(x_i - x_j)\) is the extremal coefficient and the range of \(\theta(x_i - x_j)\) is \([1 , 2]\).
Can express the F-madogram as: where \[d(x_i, x_j) = \dfrac{\theta(x_i - x_j) - 1}{2(\theta(x_i - x_j) + 1)},\] so the range of \(d(x_i, x_j)\) is \([0 , 1/6]\).
\(\checkmark\) Distance
\(?\) Algorithm
Kaufman, L. and Rousseeuw, P.J., 1990. Partitioning around medoids (PAM). Finding groups in data: an introduction to cluster analysis, pp.68-125.
Consider the \(\max \{ \| x_i - x_j \|, 2\}\) as the clustering distance.
Linkage Rule: For each pair of clusters, \(C_k\) and \(C_k'\) \[d(C_k, C_{k'}) = \frac{1}{|C_k| |C_{k'}|} \sum_{x_k \in C_k} \sum_{x_{k'} \in C_{k'}} d(x_k, x_{k'}).\]
Where can we assume a common dependence structure for extremes?
\[ \mathbb{P}(\| \mathbf{x} - \mathbf{c_k} \| < r) = 1 - \exp \left( \frac{-r^2}{2} \right)\]
Oesting, M., Schlather, M. and Friederichs, P., 2017. Statistical post-processing of forecasts for extremes using bivariate Brown-Resnick processes with an application to wind gusts. Extremes, 20(2), pp.309-332.
Perfect Prog Approach:
Simulate an ensemble from the fitted max-stable process
Assumptions:
The fitted statistical model is the truth
Relevance:
Need to ensure how we model the dependence is accurate
Create a regionalisation of Australia based on extremal dependence
Highlighted some considerations for clustering applications
Used the regionalisation to fit max-stable models
Visualised extremal dependence
Helped us understand were we can reasonably assume a single dependence structure
Non-stationary dependence!
Post-processing of compound events: Storm surge and Precipitation
t. @katerobsau
g. github.com/katerobsau